--- In [log in to unmask], Chris Bates <chris.maths_student@N...> wrote: > The following are the axioms > for a group G: No, that is the *definition* of a group, it has nothing to do with axioms. > Now the exact form a group takes may vary, but if I know something is a > group I know these things are true. If they were not, if there were some > groups where there was no identity for example, then the whole of > current group theory would just collapse. Quite to the contrary. If a mathematical construct were to violate one of the mentioned conditions, it would simply not be a group. There is no way of "knowing something is a group" other than checking whether the definition is met! The whole building of math is built on the axioms of the real numbers, which can't be proven, but must be assumed to be true for the rest of math to work. Everything else in math is proven logically on the basis of these axioms. While the integer and rational numbers are known from the Real World, the real numbers are more or less our invention. One of the axioms is the Axiom of Order. It claims that for two arbitrary real numbers x, y exactly one of the following three statements is true: x<y, x=y, x>y. This appears trivial to us, but isn't if you start from scratch, as math does. If it would turn out that the real numbers are not logically consistent, we'd be in profound feces, but then somebody would have to explain why our math worked so flawlessly for the past centuries. -- Christian Thalmann